Finite state transducer

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A finite state transducer (FST) is a finite state machine with two tapes.

Contrast this with an ordinary finite state automaton, which has a single tape. An automaton can be said to recognize a string if we view the content of its tape as input. In other words, the automaton computes a function that maps strings into the set {0,1}. Alternatively, we can say that an automaton generates strings, which means viewing its tape as an output tape. On this view, the automaton generates a formal language, which is a set of strings. The two views of automata are equivalent: the function that the automaton computes is precisely the indicator function of the set of strings it recognized. The class of languages generated by finite automata is known as the class of regular languages.

The two tapes of a transducer are typically viewed as an input tape and an output tape. On this view, a transducer is said to transduce (i.e., translate) the contents of its input tape to its output tape, by accepting a string on its input tape and generating another string on its output tape. It may do so nondeterministically and it may produce more than one output for each input string. A transducer may also produce no output for a given input string, in which case it is said to reject the input. In general, a transducer computes a relation between two formal languages. The class of relations computed by finite state transducers is known as the class of rational relations.

Finite State Transducers are typically used for morphological analysis in natural language processing research and applications.

1 Formal construction
2 Operations on finite state transducers
3 Additional properties of finite state transducers
4 See also
5 Further reading

[edit] Formal construction

Formally, a finite transducer T is a tuple (Q, Σ, Γ, I, F, δ) such that:

Q is a finite set, the set of states;
Σ is a finite set, called the input alphabet;
Γ is a finite set, called the output alphabet;
I is a subset of Q, the set of initial states;
F is a subset of Q, the set of final states; and
$\delta \subseteq Q \times (\Sigma\cup\{\epsilon\}) \times (\Gamma\cup\{\epsilon\}) \times Q$ (where ε is the empty string) is the transition relation.

We can view (Q, δ) as a labeled directed graph, known as the transition graph of T: the set of vertices is Q, and $(q,a,b,r)\in\delta$ means that there is a labeled edge going from vertex q to vertex r. We also say that a is the input label and b the output label of that edge.

Define the extended transition relation $δ *$ as the smallest set such that:

$\delta\subseteq\delta^*$ ;
$(q,\epsilon,\epsilon,q)\in\delta^*$ for all $q\in Q$ ; and
whenever $(q,x,y,r) \in \delta^*$ and $(r,a,b,s) \in \delta$ then $(q,xa,yb,s) \in \delta^*$ .

The extended transition relation is essentially the reflexive transitive closure of the transition graph that has been augmented to take edge labels into account. The elements of $δ *$ are known as paths. The edge labels of a path are obtained by concatenating the edge labels of its constituent transitions in order.

The behavior of the transducer T is the rational relation [T] defined as follows: $x [T] y$ if and only if there exists $i \in I$ and $f \in F$ such that $(i,x,y,f) \in \delta^*$ . This is to say that T transduces a string $x\in\Sigma^*$ into a string $y\in\Gamma^*$ if there exists a path from an initial state to a final state whose input label is x and whose output label is y.

[edit] Operations on finite state transducers

The following operations defined on finite automata also apply to finite transducers:

Union. Given transducers T and S, there exists a transducer $T\cup S$ such that $x[T\cup S]y$ if and only if $x [T] y$ or $x [S] y$ .

Concatenation. Given transducers T and S, there exists a transducer $T\cdot S$ such that $wx[T\cdot S]yz$ if and only if $w [T] y$ and $x [S] z$ .

Kleene closure. Given a transducer T, there exists a transducer $T *$ with the following properties: (1) $ε[T *]ε$ ; (2) if $w [T *] y$ and $x [T] z$ then $w x [T *] y z$ ; and $x [T *] y$ does not hold unless mandated by (1) or (2).

Note that there is no notion of intersection of transducers. Instead, there is an operation of composition that is specific to transducers and whose construction is similar to that of intersection of automata. Composition is defined as follows:

Given a transducer T on alphabets Σ and Γ and a transducer S on alphabets Γ and Δ, there exists a transducer $T \circ S$ on Σ and Δ such that $x[T\circ S]z$ if and only if there exists a string $y\in\Gamma^*$ such that $x [T] y$ and $y [S] z$ .

One can also project out either tape of a transducer to obtain an automaton. There are two projection functions: $π 1$ preserves the input tape, and $π 2$ preserves the output tape. The first projection, $π 1$ is defined as follows:

Given a transducer T, there exists a finite automaton $π 1 T$ such that $π 1 T$ accepts x if and only if there exists a string y for which $x [T] y$ .